Chapter 19

Lecture Presentation Software to accompany

Investment Analysis and Portfolio Management

Seventh Editionby

Frank K. Reilly & Keith C. Brown

Chapter 19

The Fundamentals of Bond ValuationThe present-value model

n

tn

p

tm i

P

i

CP

2

12)21()21(

2

Where:Pm=the current market price of the bondn = the number of years to maturityC = the annual coupon payment for bond ii = the prevailing yield to maturity for this bond issuePp=the par value of the bond

The Fundamentals of Bond Valuation

• If yield < coupon rate, bond will be priced at a premium to its par value

• If yield > coupon rate, bond will be priced at a discount to its par value

• Price-yield relationship is convex (not a straight line)

The Yield ModelThe expected yield on the bond may be

computed from the market price

Where:

i = the discount rate that will discount the cash flows to equal the current market price of the bond

n

tn

p

tm i

P

i

CP

2

12)21()21(

2

Computing Bond YieldsYield Measure PurposeNominal Yield Measures the coupon rate

Current yield Measures current income rate

Promised yield to maturity Measures expected rate of return for bond held to maturity

Promised yield to call Measures expected rate of return for bond held to first call date

Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Nominal Yield

Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

Current YieldSimilar to dividend yield for stocksImportant to income oriented investors

CY = C/Pm where: CY = the current yield on a bondC = the annual coupon payment of bond

Pm = the current market price of the bond

Promised Yield to Maturity• Widely used bond yield figure

• Assumes– Investor holds bond to maturity– All the bond’s cash flow is reinvested at the

computed yield to maturitySolve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

n

tn

p

tm i

P

i

CP

2

12)21()21(

2

Computing the Promised Yield to Maturity

Two methods

• Approximate promised yield– Easy, less accurate

• Present-value model– More involved, more accurate

Approximate Promised Yield

Coupon + Annual Straight-Line Amortization of Capital Gain or Loss

Average Investment

2

APYmp

mp

PPn

PPC

=

Present-Value Model

n

tn

p

tm i

P

i

CP

2

12)21()21(

2

Promised Yield to CallApproximation

• May be less than yield to maturity

• Reflects return to investor if bond is called and cannot be held to maturity

2mc

mc

PPnc

PPC

AYC

Where:

AYC = approximate yield to call (YTC)

Pc = call price of the bond

Pm = market price of the bond

C = annual coupon payment

nc = the number of years to first call date

Promised Yield to CallPresent-Value Method

Where:

Pm = market price of the bond


nc = number of years to first call

Pc = call price of the bond

ncc

nc

ttm i

P

i

CP

2

2

1 )1()1(

2/

Realized Yield Approximation

2

PPhp

PPC

ARYf

f

Where:

ARY = approximate realized yield to call (YTC)

Pf = estimated future selling price of the bond


hp = the number of years in holding period of the bond

Realized YieldPresent-Value Method

hp

fhp

ttm i

P

i

CP

2

2

1 )21()21(

2/

Calculating Future Bond Prices

Where:

Pf = estimated future price of the bond


n = number of years to maturity

hp = holding period of the bond in years

i = expected semiannual rate at the end of the holding period

hpn

phpn

ttf i

P

i

CP

22

22

1 )21()21(

2/

What Determines Interest Rates

• Inverse relationship with bond prices

• Forecasting interest rates

• Fundamental determinants of interest rates

i = RFR + I + RP where:

– RFR = real risk-free rate of interest

– I = expected rate of inflation

– RP = risk premium

What Determines Interest Rates• Effect of economic factors

– real growth rate– tightness or ease of capital market– expected inflation– or supply and demand of loanable funds

• Impact of bond characteristics– credit quality– term to maturity– indenture provisions– foreign bond risk including exchange rate risk and country

risk

What Determines Interest Rates

• Term structure of interest rates

• Expectations hypothesis

• Liquidity preference hypothesis

• Segmented market hypothesis

• Trading implications of the term structure

Expectations Hypothesis

• Any long-term interest rate simply represents the geometric mean of current and future one-year interest rates expected to prevail over the maturity of the issue

Liquidity Preference Theory

• Long-term securities should provide higher returns than short-term obligations because investors are willing to sacrifice some yields to invest in short-maturity obligations to avoid the higher price volatility of long-maturity bonds

Segmented-Market Hypothesis

• Different institutional investors have different maturity needs that lead them to confine their security selections to specific maturity segments

Trading Implications of the Term Structure

• Information on maturities can help you formulate yield expectations by simply observing the shape of the yield curve

Yield Spreads• Segments: government bonds, agency

bonds, and corporate bonds

• Sectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities versus BBB utilities

What Determines the Price Volatility for Bonds

Bond price change is measured as the percentage change in the price of the bond

1BPB

EPB

Where:

EPB = the ending price of the bond

BPB = the beginning price of the bond


Four Factors

1. Par value

2. Coupon

3. Years to maturity

4. Prevailing market interest rate


Five observed behaviors1. Bond prices move inversely to bond yields (interest rates)2. For a given change in yields, longer maturity bonds post larger

price changes, thus bond price volatility is directly related to maturity

3. Price volatility increases at a diminishing rate as term to maturity increases

4. Price movements resulting from equal absolute increases or decreases in yield are not symmetrical

5. Higher coupon issues show smaller percentage price fluctuation for a given change in yield, thus bond price volatility is inversely related to coupon


• The maturity effect

• The coupon effect

• The yield level effect

• Some trading strategies

The Duration Measure

• Since price volatility of a bond varies inversely with its coupon and directly with its term to maturity, it is necessary to determine the best combination of these two variables to achieve your objective

• A composite measure considering both coupon and maturity would be beneficial

The Duration Measure

Developed by Frederick R. Macaulay, 1938

Where:

t = time period in which the coupon or principal payment occurs

Ct = interest or principal payment that occurs in period t

i = yield to maturity on the bond

bond theof price

)(

)1(

)1( 1

1

1

n

tt

n

tt

t

n

tt

t CPVt

iC

itC

D

Characteristics of Duration• Duration of a bond with coupons is always less than its

term to maturity because duration gives weight to these interim payments– A zero-coupon bond’s duration equals its maturity

• There is an inverse relation between duration and coupon

• There is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity

• There is an inverse relation between YTM and duration• Sinking funds and call provisions can have a dramatic

effect on a bond’s duration

Modified Duration and Bond Price Volatility

An adjusted measure of duration can be used to approximate the price volatility of a bond

m

YTM1

durationMacaulay duration modified

Where:

m = number of payments a year

YTM = nominal YTM

Duration and Bond Price Volatility• Bond price movements will vary proportionally with

modified duration for small changes in yields

• An estimate of the percentage change in bond prices equals the change in yield time modified duration

iDP

P

mod100

Where:

P = change in price for the bond

P = beginning price for the bond

Dmod = the modified duration of the bond

i = yield change in basis points divided by 100

Trading Strategies Using Duration• Longest-duration security provides the maximum price

variation

• If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price volatility

• If you expect an increase in interest rates, reduce the average duration to minimize your price decline

• Note that the duration of your portfolio is the market-value-weighted average of the duration of the individual bonds in the portfolio

Bond Duration in Years for Bonds Yielding 6 Percent Under Different Terms

COUPON RATES

Years toMaturity 0.02 0.04 0.06 0.08

1 0.995 0.990 0.985 0.9815 4.756 4.558 4.393 4.254

10 8.891 8.169 7.662 7.28620 14.981 12.980 11.904 11.23250 19.452 17.129 16.273 15.829

100 17.567 17.232 17.120 17.064

8 17.167 17.167 17.167 17.167

Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations:

Returns to Bondholders from Naïve and Optimal Strategies," Journal of Business 44, no. 4

(October 1971): 418. Copyright 1971, University of Chicago Press.

Bond Convexity

• Equation 19.6 is a linear approximation of bond price change for small changes in market yields

YTM100 mod

DP

P

Bond Convexity

• Modified duration is a linear approximation of bond price change for small changes in market yields

• Price changes are not linear, but a curvilinear (convex) function

iDP

P

mod100

Price-Yield Relationship for Bonds• The graph of prices relative to yields is not a

straight line, but a curvilinear relationship• This can be applied to a single bond, a portfolio of

bonds, or any stream of future cash flows• The convex price-yield relationship will differ

among bonds or other cash flow streams depending on the coupon and maturity

• The convexity of the price-yield relationship declines slower as the yield increases

• Modified duration is the percentage change in price for a nominal change in yield

Limitations of Macaulay and Modified Duration

• Percentage change estimates using modified duration only are good for small-yield changes

• Difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift

• Initial assumption that cash flows from the bond are not affected by yield changes

Effective Duration• Measure of the interest rate sensitivity of an asset

• Use a pricing model to estimate the market prices surrounding a change in interest rates

Effective Duration

PS

PP

2

P- = the estimated price after a downward shift in interest ratesP+ = the estimated price after a upward shift in interest ratesP = the current priceS = the assumed shift in the term structure

Effective Duration

• Effective duration greater than maturity

• Negative effective duration

• Empirical duration

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Chapter 19